On the abstract theory of additive and multiplicative
Schwarz algorithms |
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Authors: | M Griebel P Oswald |
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Institution: | (1) Institut f\"ur Informatik, Technische Universit\"at M\"unchen, D-80290 M\"unchen, Germany , DE;(2) Institut f\"ur Angewandte Mathematik, Friedrich-Schiller-Universit\"at Jena, D-07740 Jena , Germany , DE |
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Abstract: | Summary.
In recent years, it has been shown that many modern iterative algorithms
(multigrid schemes, multilevel preconditioners, domain decomposition
methods etc.)
for solving problems resulting from the discretization
of PDEs can be
interpreted as additive (Jacobi-like) or multiplicative
(Gauss-Seidel-like) subspace correction methods. The key to their
analysis is the study of certain metric properties of the underlying
splitting of the discretization space into a sum of subspaces
and the splitting of the variational problem on into auxiliary problems on
these subspaces.
In this paper, we propose a modification of the abstract convergence
theory of the additive and multiplicative Schwarz methods, that
makes the relation to traditional iteration methods more explicit.
The analysis of the additive and multiplicative Schwarz iterations
can be carried out in almost the same spirit as in the
traditional block-matrix
situation, making convergence proofs of multilevel and domain decomposition
methods clearer, or, at least, more classical.
In addition, we present a
new bound for the convergence rate of the appropriately scaled
multiplicative Schwarz method directly in terms
of the condition number of the corresponding additive
Schwarz operator.
These results may be viewed as an appendix to the
recent surveys X], Ys].
Received February 1, 1994 / Revised version received August
1, 1994 |
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Keywords: | Mathematics Subject Classification (1991): 65F10 65F35 65N20 65N30 |
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